Critical points of multivariables

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Homework Help Overview

The problem involves finding and classifying critical points of a multivariable function, specifically f(x,y) = x^3 + 2y^3 - 3x^2 - 3y^2 - 12y. The context is within the subject area of multivariable calculus.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to find critical points by setting the partial derivatives to zero and has identified potential values for x and y. Some participants question the difficulty of remembering the conditions for critical points, while others draw parallels to single-variable functions.

Discussion Status

The discussion is ongoing, with participants providing clarifications about the process of finding critical points in multivariable functions. There is acknowledgment of the original poster's attempts and a shared understanding of the basic principles involved.

Contextual Notes

Participants note the challenges faced by students in recalling the conditions for critical points, suggesting a common experience in learning multivariable calculus.

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Homework Statement



Find and classify all critical points of
[tex]f(x,y) = x^3+2y^3-3x^2-3y^2-12y[/tex]

Homework Equations



The Attempt at a Solution



So I've gotten to
[tex]Fx = 3x^2 - 6x = 0[/tex]
[tex]Fy = 6y^2 - 6y - 12 = 0[/tex]

[tex]x=0, x=2, y=-1, y=2[/tex]

Now I can't remember how to find the critical points from here.
Any help is appreciated!
 
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The critical points are where Fx=0 AND Fy=0. Why is that hard to remember? Offhand, I would say you have four of them.
 
Oh yeah, thanks. And it's hard to remember because I'm still a student.
 
It's not that much different from how it is with functions of one variable. If y = f(x), you look for values for which f'(x) = 0. With functions of two or more variables, you look for points for which all the partial derivatives are zero.
 

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